Understanding Spivak's Proof of Unique Derivative: A Holiday Challenge

[mathlove1]

Hi--

I am trying to work through Spivak's Calculus on Manifolds over the holidays, and I am a little stuck on his proof of the unique derivative (on p. 16 as well as below).

Specifically,
(i) Why does the ≤ inequality hold, and
(ii) Why does the last equality of the second-to-last-line hold?

I would very much appreciate any help!

 

[micromass]

(i) The triangle inequality $$|x+y|\leq |x|+|y|$$

(ii) Linearity of $\lambda$ and $\mu$:

$$\frac{|\lambda(tx)-\mu(tx)|}{|tx|} = \frac{|t\lambda(x)-t\mu(x)|}{|tx|} = \frac{|t||\lambda(x)-\mu(x)|}{|t||x|} = \frac{|\lambda(x)-\mu(x)|}{|x|}$$

 

[mathlove1]

Thanks so much!